In a well-known theory of infinite series, if a series converges it is taken to represent the number to which it converges, while if a series does not converge it is not thought to represent any number. Hidden behind this well-known theory, there are some intriguing examples of the representation of numbers by series which puzzled mathematicians of the 18 c. and 19 c. When taken individually the examples might be dismissed as being distinctly odd, yet because of the substantial degree of consistency which exists between them they have been studied extensively. Early disputes about the examples centred on the question of whether there is any sense in which the series could be regarded as representing numbers. Yet those who accepted such a series and used it in their work tended to ask the simpler question, what number does this series represent?